In a way, yes
As cpcodes mentioned in the comments, the Character Creation rules say that you can do the steps of character creation in any order:
The steps of character creation are presented in a suggested order, but you can complete them in whatever order you prefer.
So you could simply take your Fighter class before your background (or ancestry, whichever is giving you the skill), allowing your background to give you free skill of your choice.
The General Rule
Generally speaking, doing the approach as you describe in your question doesn't seem to be allowed (noting the aforementioned RAW workaround of course). If for some reason the workaround wasn't possible (perhaps due to some alternate character creation rules you are using), the following is general guidance for this scenario:
While I didn't find explicit mention for it for classes, the Fighter Multiclass Dedication says this:
You become trained in your choice of Acrobatics or Athletics; if you are already trained in both of these skills, you instead become trained in a skill of your choice. (emphasis mine)
This is likely how all of these scenarios are to be handled. In that case, if you had Athletics but not Acrobatics (as a Fighter, or as someone taking this dedication), you'd have to take Acrobatics. The wording from the Class section on Initial Proficiencies is less clear, but seems to imply this as well:
If your class would make you trained in a skill you're already trained in (typically due to your background), you can select another skill to become trained in. (emphasis mine)
If you can still choose a skill you are untrained in, it doesn't quite seem to qualify as "making" you do that. It is a matter of interpretation, though, and you could probably just work with your GM to change the skill on your background if they're willing.