If two brown dwarf collide, would they become able to fuse hydrogen?

Brown dwarfs have a range of possible masses from about 13 to 80 times the mass of Jupiter. A brown dwarf has a minimum mass of 13 times the mass of Jupiter and there is a reason why this number is chosen. An object that has that mass will begin to fuse deuterium; that is why it is then called a brown dwarf instead of just an ordinary gas giant. However, just because it is fusing deuterium, doesn't mean it is fusing very much. The more you increase the mass, the higher the fusion rate you get. Once you get to about 65 times the mass of Jupiter, brown dwarfs will also begin fusing lithium. About 75-80 Jupiter masses is considered to be the upper limit of a brown dwarf, at which point much more significant fusion will be occurring and the object will be considered a true stellar object instead of just a substellar brown dwarf.

So in conclusion, any brown dwarf already is fusing hydrogen. If two collide, the resulting object will be bigger and will fuse even more hydrogen. If the total sum of their masses exceeds 75-80 Jupiter masses, then very significant amounts of hydrogen will be fusing and the resulting object will be a true star.

There is no reason in principle why not - but colliding isn't, in itself, enough. The two would of course have to merge, and that would not happen after every kind of collision. If the collision were sufficiently energetic, it could in itself generate enormous temperatures and pressures, and thus initiate fusion, but for this to be sustainable the joint mass of the pair would have to reach the usual threshold.

I think it was Carl Sagan that predicted that based on the number of brown dwarfs in our galaxy, and the established collision rate of stars, that we can estimate that at any one time in the Milky Way there are two or three main sequence stars that have formed from the collision of brown dwarfs

If the sum of their masses exceed about 80 Jupiter masses then yes.