it is known that the existence of spanning trees in arbitrary (connected) graphs implies the Axiom of Choice. I was wondering if this result still holds if we restrict ourselves to spanning trees of complete graphs. Does the Axiom of Choice still follow?
I would be very happy if you can provide a reference.
I wouldn't think so. A complete graph should always have a spanning tree.
Pick your favorite vertex $u$ from the vertex set $V$, infinite or not, and construct a giant star tree with $u$ as the center. That is, construct the tree $T = (V, F)$ where the edge set is $F = \{uv : v \in V \setminus \{u\} \}$. This has to be a spanning tree (and as pointed out in comments,note that you don't need the axiom of choice to pick an element from a single set $V$).
So denote by $S$ the statement "every complete graph has a spanning tree". If the statement "$S$ implies the axiom of choice" is true, then the axiom of choice is true since $S$ is true. So this implication cannot be - or otherwise you'd have a proof of the axiom of choice.