From the 1840s on, yes. While abolitionist white people had been helping black people escape the South for some time, the term "underground railroad" for this loose, informal network of people willing to help didn't exist until about that time. The main reason for that, of course, is that railroads were only being built for the first time in the late 1820s, so the name couldn't be lent to this other meaning before about then. The other reason is, the "Underground Railroad" is typically used to describe the network that helped people escape to Canada. Slavery was still officially legal in Ontario until the Slavery Abolition Act 1833, and while there were people who escaped to Canada before that date to avoid the United States' Fugitive Slave Act, it only became widespread in 1834, when the act of British Parliament went into effect.
Anyway, the probably apocryphal origin story for the name was first published in London in 1860 in the book The Underground Railroad from Slavery to Freedom by William M. Mitchell, an abolitionist from North Carolina who spent time in London in 1859-60 on a lecture tour (hence the location where his book was published). According to this account, a black man swam from Kentucky to Ohio across the Ohio River with the white slaveholder in hot pursuit. By the time the slaveholder had reached the Ohio side, the man had disappeared into the nearest town. When asking around where the escaped man had gone, nobody knew, and the slaveholder complained that the abolitionists in the town "must have a railroad under the ground".
The same essential story was retold in another book also called The Underground Railroad from Slavery to Freedom, though this one was published in 1898 and written by Wilbur H. Siebert. In this account, the actual man who did the escaping is named as Tice Davids, the event is said to have occurred in 1831, and it is related by a man named as Rush R. Sloane. Sloane lived in northern Ohio, in Sandusky, where Davids settled after making his escape. If true, Sloane presumably heard the story directly from Davids. The town Davids escaped to when swimming across the Ohio River is even named as Ripley, Ohio, which actually did have a known Underground Railroad depot there, the home of the rather famed abolitionist John Rankin. (He was one of the founding members of the American Anti-Slavery Society and had authored abolitionist works since the mid-1820s.) In this retelling, the slaveholder complains that Davids "must have gone off on an underground road". The abolitionists in the area, after relating the story, later changed it to a "railroad".
Siebert says that while it seems to be corroborated by the earlier Sloane story, the stories are probably only "traditions" and "they cannot be trusted in the details of time, place and occasion". While the story is possible, it could also be a deliberate narrative of somebody exalting their neighbor and a famed abolitionist to make for a good anecdote.
Either way, a figurative mention of such an escape route to Canada as a "railorad" first appeared in abolitionist newspaper The Liberator on October 11, 1839, though the article doesn't actually use the phrase "underground railroad":
"Could a great republican railroad be constructed from Mason nd Dixon's to the Canada line, upon which fugitives from slavery might come pouring into this province, at the rate of five hundred per day, its entire length would have the smile of heaven by day, and freedom's great polar lamp to guide, and the northern lights to cheer the nightly train."
The actual first use of the phrase "Underground Railroad" is thought to have been printed in the Albany Tocsin of Liberty in late September 1842:
"Tell the slaveholders that we passed twenty-six prime slaves to the land of freedom last week, and several more this week thus far. Don’t know what the end of the week will foot up. All went by the 'underground railroad'."
This is followed by messages from escaped people, to friends and family that had been left behind in the South from where the aforesaid 26 people had escaped from.
The article was reprinted in The Vermont Telegraph on October 5, 1842, and in The Liberator on October 14, 1842.
From then on, the phrase began appearing in both abolitionist press and then mainstream press with some regularity. As some examples, Vermont's Green Mountain Freeman starting using the phrase regularly in 1844, and The Anti-Slavery Bugle was doing the same by 1846. Down South, Virginia's Alexandria Gazette was writing articles about it by 1849, and a rural North Carolina newspaper was writing of the escape network by 1850. Up North in the non-abolition press, Indiana's Richmond Palladium was writing articles about it by 1848, and a rural Ohio newspaper was even making jokes using the phrase by 1849.
According to author David Blight in Passages to Freedom: The Underground Railroad in History and Memory, the phrase was well-understood no later than the mid-1840s, perhaps earlier. Harriet Beecher Stowe's best-seller and highly influential Uncle Tom's Cabin talked about people escaping on the "underground line" when published in 1852, so anybody who hadn't heard of it before that time, surely heard of it by then.
And even before the phrase was invented, people across the country knew there were abolitionists who were clandestinely (and illegally) helping escaped enslaved people get to Canada or otherwise stay undercover from slaveholders trying to recapture them. But there just wasn't a commonly-accepted name for it yet until mid-1840s.
