Remove the hard drive from your old Mac & put it in a powered external enclosure. Then use disk Utility, on your new Mac, to erase/format it & use it as a backup drive for your new Mac. Sell your old Mac, as is, for parts.
If you can't access the drive in Target Disk mode, no. External enclosure's are not all that expensive & using your old drive as a backup drive would be a smart thing to do.