No.
You should never run fsck on a mounted filesystem. Correcting errors on a live filesystem will mess up your disk. Even if you run the tool in read-only mode (without error correction) the results can't be trusted. This is true even if the filesystem is mounted read-only.
From man e2fsck:
Note that in general it is not safe to run e2fsck on mounted
filesystems. The only exception is if the -n option is specified, and
-c, -l, or -L options are not specified. However, even if it is safe
to do so, the results printed by e2fsck are not valid if the filesystem
is mounted. If e2fsck asks whether or not you should check a
filesystem which is mounted, the only correct answer is ``no''. Only
experts who really know what they are doing should consider answering
this question in any other way.
From man fsck:
For some filesystem-specific checkers, the -n option will cause the fs-specific fsck to avoid attempting to repair any problems, but simply report such problems to stdout. This is however not true for all filesystem-specific checkers. In particular, fsck.reiserfs(8) will not report any corruption if given this option. fsck.minix(8) does not support the -n option at all.
You should take the time to unmount the disk and do a proper filesystem check; results that cannot be trusted aren't useful at all.