Ok, this is my first answer on the this sub, so please help me with any formatting/missing citations. I've not formally studied this subject, but I currently work for an inter-university forum with a long history that has me researching the legacy of academic societies, forum, and colloquium. Here goes:
A large part of your question hinges on the distinction between modern conferences and the older style of learned societies that grew out of the Roman Academy. In the genealogy of academic conferences, the patriarch is Roman Academy. What made these gatherings different is the associations were not centers of learning, but rather private clubs for those already schooled in the subject to share and discuss their work (1). The two of interest to you are the Academia Secretorum Naturae founded in 1560 and it's successor the Accademia dei Lincei founded in 1603. There are a couple older societies dating as early as 1488 but they either focused on the humanities (philosophy, theology, literature, language) or have fewer accessible records such as the Sodalitas Litterarum Vistulana which I can't find a good account of (if anyone has information I would love to be pointed in the right direction) (2).
The Academia Secretorum Naturae was open to learned men who could present a new fact. This isn't actually that far off from how we make PHDs today. You have to present "new" research, often in the form of a dissertation to a panel of people already accepted as learned in your specific area of study. After a discussion/ debate (now more commonly called a thesis defense) you are either accepted or denied entry as a member of the society. The chapter on "The Renaissance and the Scientific Revolution" in A History of the Life Sciences goes into more detail. (3) Unfortunately this society was disbanded by Pope Gregory XIII.
However its successor the Accademia dei Lincei, or Lincean Academy is still around! The society was private at first, but became semi-public at the turn of the 19th century. The most famous member, Galileo, used the academy to introduce ideas and later publish some of this works. A great source on the history of the academy is David Freedberg's Eye of the Lynx (4). The practice of the scholarly society publishing for the scholars is still practiced today. Many scientific colloquium publish a book series based on talks from the previous years meetings. The Center I work for has been doing so for the past ~50 years.
I realized I gave you a lot more on the origins of the conference... and less on the content. For that I'll have to turn you over to another History of Science scholar. There are lots of archival information on what exactly the Lincean Academy was studying and who was there, but it's either in Italian or not easily accessible online.
EDIT: I'm realizing now, that I might have missed the modern part of your question and jumped straight to history. Sorry.
Dixhoorn, Arjan Van., and Susie Speakman. Sutch. The Reach of the Republic of Letters: Literary and Learned Societies in Late Medieval and Early Modern Europe. Leiden: Brill, 2008 available from Brill.com
Inventory of oldest Scholarly Societies, University of Waterloo Library - this website looks unreliable, but they really are tied to the University of Waterloo.
Magner, Lois N. A History of the Life Sciences: Third Edition, Revised and Expanded. New York: Marcel Dekker, 2002
Freedberg, David. The Eye of the Lynx: Galileo, His Friends, and the Beginnings of Modern Natural History. Chicago: U of Chicago, 2002.
5)Academia Nazionale Dei Lincei, http://www.lincei.it/index.php
Certainly not the only origin, and probably not the first, but in England at least I would nominate "The President, Council, and Fellows of the Royal Society of London for Improving Natural Knowledge" colloquially and commonly known as simply the Royal Society. Founded in 1660, it was effectively a gentleman's club with a focus on science and natural philosophy. As with many institutions in Early Modern England (and indeed, to this day), membership and invitations were heavily restricted by class. Steven Shapin's A Social History of Truth: Civility and Science in Seventeenth Century England does a good job of exploring and demonstrating the classicism, although I would caution any readers that his book is not without its flaws, most notably in overemphasizing the trustworthiness of a gentleman's word in the face of the Royal Society's stated goal of scientific proof (the Society's motto is nullius in verba - on the word of no one) and in painting the Society as much more peaceful than it was.
Members traditionally conducted or recreated experiments and gave talks, first purely to spread knowledge and their reputations, and later to actively solve problems and questions. In the second half of the 18th century, the government began putting questions to the society, which allowed for the direct application of discoveries, but also somewhat politicized the endeavors (see, for example, the rivalry between Benjamin Franklin's pointed lightning conductor and Benjamin Wilson's blunt conductor, which invariably took on a tinge of American independence politics - see Wendy Elizabeth Wagner Rescuing Science from Politics: Regulation and the Distortion of Scientific Research). It should be noted that as the Society evolved, it became less a prototype of a conference and more an organization of the type to host conferences, which it still does. The Royal Society's website also includes a brief history and a timeline, which is helpful for placing its developments in a wider historical context.
An argument can also be made for the prize schemes and competitions set up by governments to solve specific problems. The classic example of this is the Longitude Awards, created by the 1714 Longitude Act, and designed to incentivize the development of a device or method for discerning a ship's longitude while at sea. Your best source for those are the Papers of the Board of Longitude (the governing body of the awards), held (or at least digitized) by Cambridge University.
Sorry for the British focus in this answer, but that's my area of expertise. I'm also not a historian of science, so this is nowhere near as deep as it could be.
Until someone else shows up with a more pertinent answer, I figure I might help with what little information I do have. I was just reading Victoria de Grazia's amazing book Irresistible Empire on the creation of America's commercial/cultural hegemony over the West. In it, they describe the creation and goings on of the International Rotary Club-a franchise of groups for private businessmen that spread across America and Europe in the first half of the 20th century. They had weekly meetings to discuss the current and local business climate and annual conferences that took a grander approach to the same topic. So this is what I can describe to you, and from this one example we might be able to extrapolate a bit about conferences generally in the 19th and early 20th centuries.
Rotary meetings and conferences had two main objectives. The first was to act as a sort of lobbying body. Rotary clubs would support local initiatives they found favorable to their liberal business practices; women's suffrage, for example, promised to politically emancipate the most prolific consumers, women, in addition to being ethically favorable. Weekly meetings would also provide a forum to update one's peers on legislative initiatives that might concern them, rally support behind inter-sector commercial practices, and just catching up with friends and local business magnates.
At the annual international meeting, larger topics were taken up. They might get a Vice President from Ford to come and present on efficient factory practices or the CEO of a textile company to talk about free trade (a perennially lauded topic). These were also events for the most well connected businessmen from several continents to meet each other and establish contacts for future development.
De Grazia also emphasizes the Rotary Club's social aspect. It was a way for people of similar social position and role to meet one another, talk about business, but also about life generally.Due to the nature of the American economy in the early 20th century, the Rotary men were largely drawn from small businesses. As the ubiquity of conglomerates had yet to develop, the private sphere was dominated by independent enterprises vying with each other for local success and dominance. This fostered the social isolation that the Rotary Club attempted to assuage. Thus, to induce a social atmosphere, the club allowed only one member per sector per city (so only one lawyer, one tailor, one mining engineer, one pharmacist, etc). This helped to shield meetings from the harsh competition of their everyday business lives and allow for more relaxed competition to arise.
I would expect that these are also some of the main reasons for the birth of scientific conferences. By 1800, there were over a thousand scientific journals in the world, yet each of them had a limited audience due to language, distribution, etc (Kronick). While they might normally be reticent to publicize new ideas or methods, in a private setting, scientists (and indeed most people) would be more voluble. Thus, there was an urge for scientists to meet up occasionally to discuss their most recent findings, bounce ideas off of each other, and generally just be able to talk shop as in their normal lives they would find few people willing or able to have the kind of high level scientific discussions they would want.
These assemblies also provided an opportunity for scientists to assemble in large enough numbers to organize in some manner. This is where the impetus for a new journal might arise or enough members of Britain's Royal Society might band together to petition Parliament for better scientific funding.
Sorry if this was too off topic, I tried staying away from too much extrapolation of how scientists would have specifically acted, but I imagine these gatherings had a lot of similarity between them.
Kronick, History of Scientific and Technical Periodicals
De Grazia, Irresistible Empire
Coming to the thread I expected, in the West, that ecclesiastical conferences might have been oft to cover points of scientific understanding? If I could piggyback on the question I'd want to know if there were meetings within the Church - Protestant, Roman Catholic, or otherwise - which took a similar form to colloquia on natural philosophy (or other areas of science)?