Can I use a DDR5 adapter on my DDR4 motherboard?

No, you need a new motherboard — and likely a new CPU — to use DDR5 RAM. No adapter exists for desktop memory because DDR4 and DDR5 use different physical slots, voltage regulators, and signaling. Even if someone made one, the electrical differences would make it unstable or slow. Plan for the cost of a new motherboard and CPU.