Nope!
I started both trials with an egg that was nowhere close to hatching.
For the Slope Run, I attempted each difficulty multiple times, and after like 25-30 minutes, my egg was still nowhere close to hatching.
For the flying trial, regular flying doesn't affect egg hatching to begin with, so unsurprisingly, after flying around Paldea for my Gimmighoul rounds and attempting each difficulty of the flying course, even attempting hard twice, my egg still showed no sign of hatching.
Hope helped!