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

You can conduct this reaction using aryl iodide/bromide and catalyst (CuI + ligand) in presence of base {K2CO3 or K3PO4, Cs2CO3 (expensive)} in DMSO or DMF at 120 to 130 oC. Chemoselectivity could be achieved by optimizing the catalyst and reaction conditions, since there is tunable difference in the Pka value of OH and NH2 groups of aminoalcohol. The choice of ligands will be crucial. Since to make the process cost effective, I suggest you to try with ligands like proline, diammine, diketone etc. The use of pd catalyst and phosphine ligands gives better results, but you have to compromise on cost. 1,10-phenanthroline will be a little expensive choice. I dont think SNAr chemistry will work with aryl halides when you use nitrophenols.

please see ref

J. Am. Chem. Soc. 2007, 129, 3490–3491.

Angew. Chem. Int. Ed. 2009, 48, 6954–6971

Chem. Rev. 2008, 108, 3054–3131.

Chem. Rev. 2006, 106, 2651–2710.