Javascript (ES6), 151 60 bytes

New version, credits to @Leaky Nun

x=i=>console.log('x='+x+';x('+(i*(5**.5+1)/2+.5|0)+')');x(2)

Old version :

x=i=>{var s=Math.sqrt(5),a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log('x='+String(x)+';x('+f(a)+')');};x(2)

Based on this.

Javascript (ES6), 151 60 bytes

New version, credits to @Leaky Nun

x=i=>console.log('x='+x+';x('+(i*(5**.5+1)/2+.5|0)+')');x(2)

Old version :

x=i=>{var s=Math.sqrt(5),a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log('x='+String(x)+';x('+f(a)+')');};x(2)

Based on this.

Javascript (ES6), 151 bytes

x=i=>{var s=Math.sqrt(5),a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log('x='+String(x)+';x('+f(a)+')');};x(2)

EDIT: Added the quine part that my very concentrated self forgot. I thought this was too simple...

Based on this.

Javascript (ES6), 151 60 bytes

New version, credits to @Leaky Nun

x=i=>console.log('x='+x+';x('+(i*(5**.5+1)/2+.5|0)+')');x(2)

Old version :

x=i=>{var s=Math.sqrt(5),a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log('x='+String(x)+';x('+f(a)+')');};x(2)

Based on this.

Javascript (ES6), 116 bytes

var s=Math.sqrt(5),i=2,a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log(f(a))

Just change i=2 to whatever you want. This is my first attempt at golfing, so I warmly welcome suggestions because I am sure this can be further golfed !

Based on this.

Javascript (ES6), 151 bytes

x=i=>{var s=Math.sqrt(5),a=1;f=n=>{return Math.ceil((((1+s)/2)**n-((1-s)/2)**n)/s)};while(f(++a)<=i);console.log('x='+String(x)+';x('+f(a)+')');};x(2)

EDIT: Added the quine part that my very concentrated self forgot. I thought this was too simple...

Based on this.