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For example, EUR Annual (vs. 3M EURIBOR) swap has 2-Business-Days Fixing Lag. enter image description here

When interpolating with 1Y swap, the forward 3M Euribor rate starting date is the reset date（2023/6/14）, not the reset date - 2 business days（2023/6/12）.

My question is does Quantlib support custom fixing lag? If anyone could answer, I'd be very grateful.

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The helper doesn't support a custom fixing lag, but for standard quoted swaps (the ones you would use to create a curve) the fixing lag you're seeing is designed to cancel out with the index own settlement days.

To take your example: for a coupon that starts on 2023/06/14, the fixing date (giving the lag) is two business days earlier, 2023/06/12. But the Euribor 3M fixing on 2023/06/12 is calculated based on two settlement days, so the underlying forward rate starts two business days later, on 2023/06/14, and ends after 3 months. This makes the underlying rate span the same period as the coupon. The helper manages this correctly.

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