The defender went up for an aerial challenge with Lyon's Maxwell Cornet in the first half and came off worse, landing awkwardly on his back.
He received treatment on the pitch for seven minutes and was eventually taken from the field on a stretcher and in a neck brace.
Interim manager David Unsworth said in his post-match press conference that Martina was concussed and sent to hospital for scans.
He said: "Cuco was concussed and complaining of pains down his neck. He's gone straight to hospital for X-rays and scans. We wish him all the best."
Martina's injury compounded a bad day for Everton, as the 3-0 loss confirmed their elimination from the UEFA Europa League.