How to obtain as best as possible video quality in PSX/PS2 on modern TV
Solution 1:
It's unlikely that converting the composite signal to HDMI will result in a significantly better quality than wiring it directly to the TV. The cheap converters are generally not much better than what your TV already has built in, and while a Framemeister-class upscaler might wring out some additional quality, you will ultimately be limited by the composite signal.
Unfortunately, even using the PS1's A/V multiout is not a guarantee that the quality will be any better. A lot of cheap converters simply use the composite signal, which will give you the same or worse quality as simply using the composite output of your PS1 (and potentially additional headaches, if your TV does not support 240p over HDMI). Even converting from SCART or S-Video does not guarantee an improvement in quality, as both of those standards allow composite video to be carried.
To be certain that the picture quality will improve over simply using the composite output, you'll need a proper upscaler, such as an OSSC or a RetroTink 2X-Scart.
Or, if you have a PS2, you might be able to use a YPbPr component cable and play your PS1 games using backward compatibility - some modern TV sets support that standard, and some of them will also accept 240p signal over component even if they have trouble accepting it over HDMI. Unfortunately, this is not an option for the PS1, which does not allow switching between RGB and YPbPr component.
Solution 2:
The best way to get the highest possible quality is to use one of those Digital to Digital Converters. They are mods installed inside of the console that take the digital video signal before it goes to the conversion for RCA/SCART (DAC).
For the PS1, there is the PS1Digital. Is a quite expensive kit that costs about 170 USD but it allows you to get a native HDMI output from a PS1. Here is a video showing how the PS1Digital works.
For the PS2, there is still not a native HDMI solution but the developers of the DCHDMI (HDMI for the Dreamcast) are working on an HDMI solution for the PS2. Until that releases, your best bet would be high quality component cables and the Open Source Scanline Converter.