What is your X? The phenol you are talking about is the aminophenol? 

There are a few ways to do this. SNAr or Buchwald type coupling reaction.

Either way I think you need to either protect the amino group or use a nitro group instead. The amino is likely to be competing with the phenol

If I may suggest: change your reagent with nitrophenol (very cheap and more acid), make the ether coupling reaction with your substrate and then reduce the nitro to amine under very mild conditions.

You can try mitsunobu reaction using DEAD or DIAD with PPh3.

I agree with Matteo, I would even consider using the 4-fluoro nitrobenzene (CAS: 350-46-9) and prepare your ether via an SNAr, your quinolinol (X=OH) being the nucleophile. You could then reduce of the nitro using standard  conditions (Fe/acid, H2/Pd/C...).

I would recommend You another way to solve a problem, If You use K2CO3 as a base, water apperars in the reaction mixture and return your phenole, more, not all the aminiphenol transform into phenolate. So if You use a metododlgy such as.... to a solution of aminophenole in dry DMF add 1.0-1.1 eq. of NaH and stirr until gas evolution finished, you will obtain DRY and pure aminophenolale that You can introduce into a subsequent reaction with quiniline with leaving grouhs such OMes or OP(O)OR

May I suggest saccharin chloride or 5-chloro-1-pnenyltetrazole as derivative ether agents. Then try your substitution reaction. Good luck