Observation of the muon spin precession in high magnetic fields imposes strict requirements on the time resolution of scintillation counters used for detection of incoming muons and the decay positrons. The time resolution of PMT-based detectors is limited in high magnetic fields due to an additional broadening of the light pulses in the necessary light guides. The detector systems using photodetectors that are insensitive to magnetic field, such as multipixel Geiger-mode avalanche photodiodes (G-APDs), do not require light guides and thus can prove to be more advantageous for such application. In this work we build prototypes of fast-timing muon and positrons counters based on G-APDs. The detectors were tested in magnetic fields up to 4.8 T and showed a time resolution better than 50 ps (sigma).