It sounds like it could be a digital station on the same frequency EMS uses for paging. Does the noise sound something like
this MotoTRBO sample, or
this P25 sample? If so, that is probably the case. The reason you would only hear it after the radio alerts is because under certain circumstances, the radio enters monitor mode after receiving a page. If EMS uses PL/DPL, and their radio alerts for pages but still hears other traffic on their channel (signalling squelch is set to "Or"), then the radio will enter monitor mode after receiving a page.
The easiest way to stop it is to decrease the Auto Reset Timer to 1 (the minimum), and set the Auto Reset Type to "Auto w/o Carrier Override". Although if they have a paging only channel (with signalling squelch set to "And"), doing this will of course screw that up and they will no longer hear voice traffic after the radio alerts on that channel. Another option would be to disable page decoding altogether. All of this must be done using the programming software.
In the mean time, if this is indeed what's happening, the only thing you can do is go to the radio and press the monitor button (if programmed), or change to a different channel then back again after the radio alerts. Either of these will reset the muting conditions.