I’m making a drow, and part of his backstory is that he learns to worship another goddess; the platinum dragon Bahamut. You have to be L/G to worship him, but Lolth is still a part of drow life, so could he worship both without being attacked by her or her followers? (It doesn’t really matter but this is 4e)
She drove the drow into heavy infighting under the pretense of culling the weak, while her real goals were to hold absolute control over the dark elves, prevent the rise of alternative faiths or ideas, and avoid complacency (even though she found amusement in the strife that plagued her followers' communities).
so: (1) no, generally speaking Lolth would not be tolerant of your character's worship of a different god; but (2) your character is still likely to be attacked by Lolth's followers, just out of general spite, even if your character worships Lolth exclusively.
Lolth is legendarily jealous.
As in "once upon a time she betrayed the gods and stole the drow for her personal use". Assuming your DM's running the default 4E campaign setting they put the books out for, and not, say, the Forgotten Realms, which has its own wrinkles. Here's a fun little bit from Underdark, p.73:
The drow are all but monotheistic, recognizing no other divine beings as true gods. Bahamut, Moradin, Torog, and the rest are acknowledged as real, and powerful, but as beings of lesser status. Those caught praying to or drawing divine power from false gods are consigned to the Fane [of Lolth] as sacrifices.
Of course, that's only true for the drow who count themselves part of proper drow society and its endless games of spider-and-also-spider. An outcast can fall in with any god, because what are they going to do, kill you twice?