Python 3, 87 bytes

import math
a=4;b=2
while 1:math.gcd(a,b)>1==print(f'{a-b}\n{b}');a+=b<3;b=[b,a][b<3]-1

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Here, output is like

2
2
2
4
3
3

Python 3, 79 bytes

import math
a=4;b=2
while 1:math.gcd(a,b)>1==print(a-b,b);a+=b<3;b=[b,a][b<3]-1

Try it online!

Here, output is like

2 2
2 4
3 3

APL(Dyalog Unicode), 29 28 bytes ^SBCS

-1 byte by replacing () with an ¨ on ,¨∘⌽⍨∘⍸

Prints the first n pairs.

{⍵↑⊃,⌿,¨∘⌽⍨∘⍸¨1≠∨∘⌽⍨¨⍳¨⍳⍵+9}

Try it on APLgolf!

Charcoal, 39 bytes

ＮθＷ‹Ｌυθ«→ＦΦ…²ⅈ⊙…²ⅈ¬∨﹪ⅈμ﹪κμ⊞⊞Ｏυκ⁻ⅈκ»Ｉ…υθ

Try it online! Link is to verbose version of code. Takes n as input and outputs the first n elements. Explanation:

Ｎθ

Input n.

Ｗ‹Ｌυθ«

Repeat until n elements have been found.

→

Increment i+j.

ＦΦ…²ⅈ⊙…²ⅈ¬∨﹪ⅈμ﹪κμ

Loop through all values of i from 2 to i+j exclusive, for each value looping again though the same range, checking to see whether both i and i+j are divisible by the same nontrivial factor.

⊞⊞Ｏυκ⁻ⅈκ

For those that have a common factor, push i and i+j-i=j to the list of found elements.

»Ｉ…υθ

Output the first n elements.

Jelly, 14 bytes

żṚṄ€g/Ḋ$¡€ṛŻ‘ß

A full program that prints the sequence values forever.

Try it online!

How?

żṚṄ€g/Ḋ$¡€ṛŻ‘ß - Main Link: no arguments (X = implicit integer zero)
 Ṛ             - reverse {X} (0 -> [0])
ż              - {zip X} with {that}
         €     - for each pair:
        ¡      -   if...
       $       -   ...condition: last two links as a monad - f(pair):
     /         -                   reduce by:
    g          -                     greatest common divisor
      Ḋ        -                   dequeue -> [2..that] (the first pass has pair [0,0] with GCD 0, so can't just decrement as -1 is truthy - but it saves a byte or two elsewhere)
  Ṅ€           -   ...then: print each on its own line
           Ż   - zero range (first pass -> [0]) / prefix with 0 (otherwise)
          ṛ    - get the right argument -> {that}
            ‘  - increment (vectorises)
             ß - call this link again with X set to {that}

05AB1E, 19 bytes

∞εL¨Ryδ‚ʒ¿≠}εηíÆR€,

Port of @Arnauld's JavaScript answer, so make sure to upvote that answer as well!
Also outputs the infinite sequence on separated newlines.

Try it online.
Try it online with more readable format.

Explanation:

∞          # Push an infinite positive list: [1,2,3,...]
 ε         # Map each value to:
  L        #  Convert it to a list in the range [1,value]
   ¨R      #  Remove the last item and reverse it: range (value,1]
      δ    #  Map over each inner number
     y ‚   #   Pair it with the current value
  ʒ        #  Filter this list of pairs by:
   ¿       #   Check the pair's GCD (Greatest Common Divisor)
    ≠      #   Check that this is NOT 1
  }ε       #  After the filter: map over each pair [b,a]:
    η      #   Get its prefixes: [[b],[b,a]]
     í     #   Reverse each: [[b],[a,b]]
      Æ    #   Reduce each by subtracting: [b,a-b]
       R   #   Reverse the pair: [a-b,b]
        €  #   Map over both values:
         , #    Output it with trailing newline

JavaScript (ES6), 59 bytes

-13 thanks to @tsh!

Prints the sequence indefinitely.

for(a=b=c=1;;)a%c+b%c--||(c&&print(a-b+`
`+b),c=b=b-1||a++)

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More readable output