I have an Apple computer connected to an HP printer and it is printing every other page with a blank page between

I see an OfficeJet 4500 G510, and an OfficeJet 5100 series and an OfficeJet 5510 series, but do not immediately see any OfficeJet 510 printer. HP indicates that the 4500 G510 and the 5510 are compatible with OS X 10.10, but does not list any OfficeJet 510 series.

First, check that you have the current HP printer drivers loaded, then ensure that the printer has the current firmware loaded, and then consider opening a support case over at HP.

I've had success with Brother MFC-series laser printers with macOS, as an alternative to the HP printers.

You could check with the HP printer support folks, asking them for the printer-local settings and any specific macOS settings they recommend for the particular HP printer and macOS software version involved. They're usually familiar with these sorts of setup and configuration glitches, too.

For somebody here to look at the docs and related, we'll need to know the macOS version in use, and particularly the model of HP printer, as well as whether the particular HP printer is connected to a wired or Wi-Fi network, or if the printer is directly connected to the Mac via USB or some other local cabling. (HP has offered many printers over the years, and with different cabling, networking, setup, menus and options.)

You might be able to find a driver here.

Gimp - Print

HPIJS for Mac OS X

PrintFab