You can customize your QR code using the Instagram app for Android and iPhone, but not from a web browser.
To customize your QR code in the Instagram app for Android and iPhone:

Instagram app for Android and iPhone:
Tap

or your profile picture in the bottom right to go to your profile.
- Tap
, then tap QR code. - To customize it, touch anywhere on the screen or tap the button at the top to try other designs.
- Use Color to switch between different colors, Emoji to choose an emoji design or Selfie to take a photo and try on different selfie stickers.