PCB Mount Relay Causing Reset
First, check to make sure the flyback catch diode D18 is really there, not broken, and the connection intact.  Without it, this is exactly the kind of symptom you'd expect, in addition to Q3 eventually getting fried.

Second, make sure the PIC is properly bypassed and that MCLR isn't floating.  Make sure there is a 1 &micro;F or so bypass cap from every power pin directly to the nearest ground pin.  Note that power pins include AVdd, not just Vdd.  All the Vss pins must be tied together solidly right under the PIC.

Otherwise, this is probably due to noise caused by the switching itself.  The relay contacts and what is connected to them should be kept away from the low voltage circuitry.  Proper grounding of your circuit will also minimize the problems caused by nearby transients.