The injury was scripted rather than real
Leonard Nimoy - Often asked: Why did Spock smile and limp in the first ST pilot? Answer: Was told to smile by the director/ to limp
by Gene Roddenberry.
As to why Spock was limping;
Trivia: In case you were wondering why Spock is limping in the scenes
on Talos IV, the original script for “The Cage” indicated that he’d
been injured on Rigel VII, in the fight that Pike briefly discusses
with Dr. Boyce.
Star Trek Re-watch: “The Menagerie” Part I
This 74-page script is very narrative heavy--everything is described
and explained. Again, very understandable, it was setting the scene,
describing the Trek universe for the first time.
It is noted that Spock's limp and the bandages on the various
crewmembers are a result of the recent battle on Rigel VII, thus
laying to rest all the speculation that Leonard Nimoy was limping
because of a real life accident.
The Cage: Revised Draft
From the episode itself;
BOYCE: Sometimes a man'll tell his bartender things he'll never tell his doctor. What's been on your mind, Chris, the fight on Rigel
PIKE: Shouldn't it be? My only yeoman and two others dead, seven injured.
The Menagerie: Transcript