Quinine will not under go radical polymerization at that double bond, the radical generated is too unstable. If you look at styrene or methylmethacrylate you can see how the radical generated is stabilized by delocalization and/or sterics. You have two routes you can pursue to make a quinine functionalized polymer, neither of which is guaranteed to maintain the exact biological activity of quinine (which I what I'm assuming you want). The first is to use that alkene or the alcohol to couple quinine to a simple monomer (such as reacting the alcohol of the quinine with methacroyl chloride). The other would be to do postpolymerization modification. In that case, you could take a simple monomer like 4-vinylbenzyl chloride, and (attempt) to react it with again, the hydroxy on quinine, which to me is the simplest handle on the molecule. You could also use an activated ester functionality like NHS or PFPA to make this reaction easier, if more synthetically involved up front to make those ester monomers.
Another handle could be to try what would be a longer synthesis and do hydrosilation with that double bond, or some sort of olefin metathesis. Theres room to do other things, but these are the standard ways to make polymers with exotic functionalities.