Yes, they did! They called them "paper darts" since the term "airplane" didn't exist yet.

We don't know exactly when they became a children's toy, but in 1799 George Cayley published one of the first treatises about aerodynamics from the modern perspective, and he built several small gliders from wood and cardboard to test his theories.

From around that same period, Charles Matthews, an English actor and comedian of the early 19th century, recounts in his memoirs that when he was a boy he and his companions often "shot paper darts" trying to get them to "sail and stick in the headmasters wig"! These incidents took place in around 1786 when Matthews was a student at the Merchant Taylors' School, which was at that time located in the City of London.

Finally, one of the most direct evidences of the existence of paper airplanes or paper darts is from materials discovered in 2012 during the renovation of a old chapel and former school building in Devon: http://www.dailymail.co.uk/news/article-2111963/Schoolboys-100-year-old-paper-planes-pre-date-manned-flight-eaves-chapel.html

While it is the Daily Mail (aka the Daily Fail, if I may editorialize for a moment), the photos do show small delta-winged dart-like toys made of paper and the noses weighted with what appear to be parts of metal pen nibs. The other papers in the building date these artifacts to the late 19th century, maybe the 1880s or 1890s. If some scholar should want to undertake a further study of the subject, I believe the paper darts from St. Annes are today in the collection of the Museum of Barnstaple and North Devon.

Paper airplanes are something of a hobby of mine, as you might have guessed.

There is an answer to a similar question here by /u/cthulhushrugged.