Google Sheets, 53 bytes

=match(abs(A2),sequence(abs(A2)+abs(B2),1,,abs(B2)),)

Outputs a positive integer when the rational is an integer, and an error value otherwise.

Retina 0.8.2, 21 bytes

-?\d+
$*
r`^\1*/(1+)$

Try it online! Link includes test cases. Explanation: Converts to unary, then tries to find a multiple of the absolute denominator that equals the numerator.

If that looks too much like division, then for 27 bytes:

-?\d+
$*
+`(1+)/\1$
/$1
^/.

Try it online! Link is to verbose version of code. Explanation: Converts to unary, then subtracts the absolute denominator from the absolute numerator as many times as possible, then checks to see whether the result is zero. (Removing the trailing . would allow 0/0.)

Vyxal 3 -l, 6 (literate) bytes

both: (: abs factors ) subset? all

Vyxal It Online!

Takes denominator then numerator

Uiua 0.15.0-dev.1, 8 bytes ^SBCS

∊×⇡+1,∩⌵

Try on Uiua Pad!

Takes n m and outputs 1 iff \$\frac{m}{n} \in \mathbb{Z}\$ (takes the stance that 0/0 = 1, will work on finding a solution that does not)

Explanation

∊×⇡+1,∩⌵
      ∩⌵ # absolute value of both
  ⇡+1,   # range [0 ... m]
 ×       # times n
∊        # is m in this list?