This is a problem that will be happening more and more now with newer washing machines and the newer requirements for them to be on GFCIs. Newer designs of washers are now using variable speed electronic motor controls inside, rather than the old simpler plain induction motors and gear boxes. The Variable Speed Drives that are inside are notoriously difficult to keep GFCIs from holding in consistently, because they create a lot of what is called "Common Mode Electrical Noise", which the GFCI circuitry sometimes interprets as leakage to ground.
One simple thing you can do is to double check ALL of the grounding (green) wires in/on the washing machine frame, make sure nothing is loose or corroded. If that doesn't help your choices are to live with it, or try installing different brands of GFCIs, because this issue seems to follow some brands more than others. If your breaker is Square D, that's one that is notorious for this issue. You can't usually install another brand of breaker in a Square D panel though, but the other option is to use a standard breaker, then put in a GFCI receptacle at the outlet. Then if that receptacle nuisance trips, keep changing brands until you find one that doesn't.