We are pretty sure.

Mozart and Beethoven really ran the same circles, especially when you consider Mozart's close friendship with Haydn, and how Beethoven spent a good bit of his life learning from Haydn when he was young (20s?), plus each composer's affinity to the cultural capital at the time, Vienna.

Beethoven, at a young age, spent quite a bit of time visiting Vienna, notably in 1787 and 1792, where he soon realized Vienna was the city for him, and Mozart had been living in Vienna since 1781.

The story goes that in April, 1787 Mozart and his wife Constanze heard young Beethoven playing one of his pieces. Constanze said something along the lines to Mozart that he should remember Beethoven's name for it will be heard often (source needed).

Beethoven finally moved into Vienna, but at this time Mozart had already been dead for nearly a year.

It's important to note that Beethoven was sixteen when he met Mozart, and Mozart was near the end of his life (died in 1791). Beethoven didn't really come into his own until his early to mid-20s. Mozart too was just about to hit the low of his life, when his friends would leave Vienna, and his wife gone, recovering from illness, his father and him with little to no relationship, and his mother long dead.

I'm not a music historian per se, but I guess I'll get the ball rolling. My music history professor in college had his own ideas about what happened here, but I think this is the generally agreed upon info:

They were (briefly) in Vienna at the same time in the beginning of 1787.. Beethoven ended up leaving Vienna prematurely to attend to his dying mother. By the time he returned to Vienna several years later, Mozart had passed away. It's hard to say whether or not they actually met, but it's safe to say that Beethoven would have liked to meet/study from Mozart.

There are no surviving documents relating to their meeting. There are second-hand anecdotes, of uncertain reliability. Thayer's Life of Beethoven discusses this. Cooper and Skowroneck agree, and also cite Thayer.