The freedom of movement spell only prevents difficult terrain from affecting our movement. The saving throw happens regardless. The effects of failing the save are somewhat ambiguous
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 our movement. Note: it does not provide immunity to other effects of difficult terrain, such as changes to our speed, hit points, or anything else.
Spells and magic effects cannot reduce our speed.
Spells and magic effects cannot make us 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 their 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. The feature never says that certain creatures are immune to the save, so any creature that starts its turn in the area will make the saving throw.
The terrain is still considered difficult terrain, and freedom of movement only prevents it from affecting our movement. Simply making a saving throw is not affecting our movement so we would make the save no matter what. If we were to fail the save, there are two possibilities:
Failing the save does not count as difficult terrain affecting our movement, instead the Gibbering Mouther itself is affecting our speed; our speed, and thus movement, will both reduce to zero.
Failing the save counts as the difficult terrain affecting our movement through affecting our speed; our speed will reduce to zero but our movement will remain the same. This is a strange, though not inherently impossible, situation.
This odd situation of having 0 speed but still being able to spend movement also shows up when grappled, as discussed in the question "Is Freedom of Movement hampered from actually working to escape a grapple?". Though how the scenario resolves is different (freedom of movement explicitly allows us to spend movement to escape a grapple), it shows that the idea isn't impossible.
There are also spells like spike growth, which create a damaging area of difficult terrain; freedom of movement will not somehow prevent us from taking damage; it only prevents the area from affecting our movement. This further supports that freedom of movement does not prevent all effects of difficult terrain, only ones which affect our movement.
In the end, we will make the save no matter what as making a save has no effect on our movement and freedom of movement only prevents difficult terrain from affecting our movement, nothing else. That said, what happens when we fail the saving throw depends on whether a GM believes it is the difficult terrain or the Gibbering Mouther that caused us to make the save.
Nothing in the description links the terrain to the save; however, the feature is called "Aberrant Ground" so I can see arguments made both ways.