The Beosystem 1 only as an analog VGA output (it dates back to the late 90s). The Beovision 4-65 does have a DVI-D connection, but that could only be used with a Beosystem 3 (and maybe a Beosystem 4). The Beovision 4-65 should also have an analog VGA input; this is the input you would use to connect to the VGA out of the Beosystem 1.
Further, note that to work with the Beovision 4-65, the Beosystem 1 would need to be of a certain minimum software level, and would have to have the proper setting in the "secret" service menu. In addition to the the VGA connection, you need a control connection between the Beosystem 1 and the Beovision 4-65; the cable of which is USB-B on the Beosystem 1 side and a DB9 RS232 connection on the Beovision 4-65 side.
Note that the Beosystem 1 only has a single 1080I (not 1080P or 720P) analog component input. this can be expanded to additional 1080I (not 1080P or 720P) analog component inputs using a Zektor component switcher (if you can find one...I probably have an extra). There are devices out there that can convert HDMI to an 1080I analog component signal (see https://www.hdfury.com).
All that said, the Beosystem 1 is a rather large dated box originally designed for the US market CRT Avant, that while fantastic for its time, is probably best replaced. On the audio front, note that it lacks any type of base management. I would encourage you to at least consider a Beosystem 3, which would provide a reasonably modern inference and user experience. Also, while I'm not sure if a Beosystem 4 would be compatible, but it combined with a Beovision 4-65 would give you an superb modern TV experience.