Can we band pass filter QPSK signal?
The matched filter is precisely that - it will filter out as much noise as is possible. 

The reason you might want to use a band-pass filter (or ideally, a root-raised cosine) is to use less bandwidth. If you look at the spectrum of unfiltered QPSK, it will have a lot of out-of-band power. You don't want this in practice (because people transmitting on those other bands will be disturbed by this) and so you filter it.