If the victim is aware of the attack, then I'd use the combat system, and either require the victim to be pinned down, or to need to use it like a cheap-quality small dagger with a -2 improvised weapon skill penalty that does no damage itself, can't pierce any armor stronger than soft leather, and does thrust-3 for purposes of piercing armor with DR1. Anything but a successful attack would likely have a good chance to mess up the syringe one way or another. Also I'd require a separate roll to inject after the stab is done, on something like the lower of close combat DX or First Aid. I'd also research to see if there would be a big difference between injecting into a vein and into muscle - if there is, then that should require a target penalty to get a vein during combat.
If you can try to inject without the victim knowing (using, say stealth versus perception and being able to come up from behind, or a ruse using Acting and Sleight of Hand versus maybe IQ/Skepticism and perception), then it might just be a matter of First Aid skill at a penalty (adjusted by the GM according to the situation - clothing, how distracted is the target, what the posture of the people is like) to do a proper injection in the right spot before the target reacts.
Once injected, if we accept the description, that sounds like an increasingly difficult HT roll each turn to avoid physical stun and falling unconscious, or possibly inevitable unconsciousness after a number of seconds.
(P.S. As @JohnDallman pointed out in comments, if you want to track a more accurate medical skill for injections than First Aid, that might make sense. I chose First Aid because it seems like the closest GURPS skill I actually give characters other than medical specialists, and of course First Aid defaults to Physician, but clearly there could be people who were skilled in injections but not the rest of First Aid, which could be a Specialized skill or One-Trick-Wonder as @JohnDallman suggested.)
(P.P.S. Thinking about syringes in combat a bit more, I agree with @philosophyguy's that one could probably use close combat DX or brawling (or karate) for stabbing someone with a syringe in combat. But as with knife, I would have a skill penalty and any miss or parry may mess up the syringe. That represents that while you might be able to poke someone with it at full DX/brawiling/knife skill, you have to do it carefully or the syringe may be broken or emptied before it can inject.)