Install Canon MG5300 printer

You are entirely at the mercy of Canon to provide compatible drivers for Monterey. It would appear that Canon last provided drivers for this printer with macOS High Sierra (10.13.*).

However, Canon did provide an AirPrint firmware update that would upgrade your printer to firmware version 2.020. A USB cable connection is necessary for that update, and if your printer firmware is already at this version, you do not need it. See this link. You will need to click Software and Drivers, and then select macOS 12 for the operating system, as I had to do here:

Although Apple does not show the MG5300 as AirPrint compatible, that was likely based on the older firmware revision, and once your firmware is up to 2.020, and the device is networked (wired/wireless - but not USB), Apple's built-in Airprint should recognize the device and support printing without Canon drivers.

Once you have the firmware update v2.020 applied, power off the printer, configure it with either an Ethernet cable, or wireless, and with it connected to your Mac power it on again. In System Preferences > Printers & Scanners if the printer appears in your Printers window, click [-] to remove it. If no printer appears in the previous, then click [+]. In either case, this will initiate a Bonjour search for your printer, and should show a line item that it was found. Click that, and hopefully, the Use selector will auto-select and give you AirPrint (or default). Click Add to re-add the printer and try a test print.

Hi,

The following models support the driverless AirPrint technology:

About AirPrint - Apple Support

• Canon MG5310
• Canon MG5320
• Canon MG5330
• Canon MG5340
• Canon MG5350
• Canon MG5370
• Canon MG5380

Is you printer one of these?

If so, an AirPrint-enabled model is compatible with macOS, and can be used without additional drivers, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).

Add a printer to your printer list so you can use it on Mac – Apple Support (UK)

Reset the printing system on your Mac to solve a problem – Apple Support (UK)

This guide from Canon Europe could perhaps be of interest to you:

https://www.canon-europe.com/support/consumer_products/apple_airprint_guide/#Pmac

If there is a problem with AirPrint discovery for one or more devices, and all settings are OK, you could try the following full procedure (in this order):

Power off the Wi-Fi router, the printer, the Mac computer, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the Mac computer. Finally, power on all other devices/computers.

You could also test the AirPrint capabilities via an iPhone/iPad.

Use AirPrint to print from your iPhone, iPad, or iPod touch - Apple Support

EDIT: Just noticed that VikingOSX had posted a reply while I was composing my message.