Big Sur does not print on Laserjet4Plus

Apple does not install the HP Laserjet 4 Plus driver. That is your responsibility, provided one compatible with Big Sur can be found.

Since you have an Intel iMac, and on Big Sur, you will find a Laserjet 4Plus driver in the HP 5.1.1 Printer Software Update. This installation package is not supported on macOS Monterey (macOS 12) and later, and thus, is the last support on macOS that you will find for this printer model since it is not supported by Apple's AirPrint, or by HP.

The preceding installation will not magically enable printing. You need to ensure that the printer is powered, and in some manner (preferably networked) is connected to your iMac. Then visit System Preferences > Printers & Scanners panel. If there is already an entry for the Laserjet 4Plus, then right-click on it and choose Reset printing system… If there is no entry present, then click the [+] button to add it. If it is connected via the network, the next panel will show that Apple's Bonjour service found it. Click that line item and then in the Use selector, select the HP LaserJet 4Plus driver entry, and then click Add. Once this is done, you should be able to print.

If you have a USB-cable between your printer and your Mac then it will behave that way. If there is no Ethernet jack on the back of the LaserJet 4Plus, or if there is without a cable running from your printer to a network switch, then you don't have an Ethernet solution.

Since you have the HP driver software installed, you need to Reset your Print system, and on a subsequent panel, you need to click the Use selector and specifically pick that HP Laserjet 4Plus postscript driver from the list of available HP printer drivers. Then click Add.

See Reset the printing system on your Mac to solve a problem - Apple Support

I have never owned a USB printer. My first printer that I purchased in 1994 was a LaserJet 4M Plus that had an Ethernet port, AppleTalk/LocalTalk, and a Parallel port. It lasted until 2017.

Hello VikingOSX,

The HP LaserJet 4 Plus (C2037A) should be more or less the same printer as the HP LaserJet 4M Plus (C2039A), with a few exceptions:

An HP LJ 4 Plus would support enhanced PCL5 only, whereas the HP LJ 4M Plus supports enhanced PCL 5 and PostScript level 2. A PostScript upgrade appears to have been available for the HP LJ 4 Plus.

The LJ 4 Plus ports would be parallel and serial, where the LJ 4M Plus would have parallel, serial, and LocalTalk ports.

Ethernet for the LJ 4 Plus would require a (separate) print server (could be connected to the parallel port).

If maxfromcayucos is using USB, it would probably be via a parallel-to-USB adapter.

After a printing system reset, as you suggested, it ought to work with the HP LaserJet 4 Plus driver (or one of the other HP LaserJet 19.13 drivers) listed in Printer and scanner drivers for Mac - Apple Support. If necessary, a test could be made with a generic PCL driver as well.

hp do not seem to support that printer model on newer than  Mac OS X 10.5 Leopard 

HP LaserJet 4 Plus/m Plus Printer series Software and Driver Downloads | HP® Customer Support

Is this a networked printer, or connected by other means? Just installing the HP driver package does not solve the printing problem. You need to Reset the print system and explicitly tell the printer to use the appropriate HP driver, then click Add to associate the printer with that driver.

It should just be a package PKG or DMG installer in your Downloads folder that you double-click and when prompted during the installation process you supply your administrator (login) password to complete it.