Do I need a paid plan to customize my QR code?

No. The QR Code Designer is available on every plan, including the free tier. You can pick a style preset, set colours, upload a logo, and download as PNG or SVG without upgrading.

What differs between plans is the click limit on the short link behind the QR. Every scan counts as a click, so a busy QR can eventually exceed the free tier's quota. See What happens if I exhaust my click limit? for how that works, and the our pricing for the specific thresholds.