If you buy an av analogue to hdmi converter will be a hit and miss with image quality and controller input lag.
http/www.nintendolife.com/news/2018/04/hardware_review_gamecube_hdmi_adapter_-_gc_video_plug_n_play_3_0
If your GC unit supports this plug and play adapter look into it. This adapter allows digital av to Hdmi, so no conversion and no input lag. Bit pricey albeit an adapter like no other.
I've had my N64 modded internally with Hdmi in Australia, was well worth it but had to send the unit via mail to the person was the only niggling fear of it being damaged by person or transport. All worked out and good job done and hdmi had no washed out colours as with av.
Could try an cheap av to hdmi active converter, will need to be powered by tv's usb port if have one, or from power socket.