The freedom of movement spell only prevents difficult terrain from affecting our movement. The saving throw happens regardless
The freedom of movement spell states:
For the duration, the target's movement is unaffected by difficult terrain, and spells and other magical effects can neither reduce the target's speed nor cause the target to be paralyzed or restrained.
From that we can conclude what things the spell does:
Difficult terrain does not affect your movement. Note, this says nothing about your speed or anything else difficult terrain might cause.
Spells and magic effects cannot reduce your speed.
Spells and magic effects cannot make you paralyzed nor restrained.
The Gibbering Mouther's Aberrant Ground feature is not magical, so the latter two points do not apply. What the feature does do is this:
The ground in a 10-foot radius around the mouther is doughlike difficult terrain. Each creature that starts its turn in that area must succeed on a DC 10 Strength saving throw or have its speed reduced to 0 until the start of its next turn.
Ground within a 10-foot radius becomes difficult terrain.
If a creature starts its turn within a 10-foot radius, it must make a saving throw; failing this saving throw makes your speed become 0.
The feature never says that being immune to the normal effect of difficult terrain makes us automatically succeed on (or not have to make) the saving throw. Since freedom of movement only helps prevent changes to our movement but this is a change to our speed, it will still apply to us as normal.
Similarly, with a spell like spike growth, which creates a damaging area of difficult terrain,freedom of movement will not somehow prevent us from taking damage; it only prevents the area from costing additional movement.