I was totally confused

i was totally confused. When using a microstripe whose Z0^2 equals Zin*Zl and length is quarter of the wave legnth to match the Zin and Zl, the gamma_in should be zero. But gamma_in equals (Zin-Z0)/(Zin +Z0), while gamma_in won't be zero.
e.g. Zin=50 Zout=75 then Z0 should be 61.24 while gamma_in equals (75-61.24)/(75+61.24)

Hi

you are talking about a quarter-wave transformer.

what u cal : Zi^2 = Z0*Zrequired. Here Zi is the impedence of the TL u must use for the transformer to do impedence matching. U then go ahead and insert 1/4 wavelength (at the operating freq) of this TL .

Look up any text that teaches smith-chart, they'll explain it better.

But shouldn't it be normalized by the characteristic impedance of th TL?

When under matched condition, Zin will see an impedance equal to Zin itself (that is the meaning of matching). Hence the gamma is zero.

The gamma_in equals (Zin-Z0)/(Zin +Z0), while gamma_in won't be zero in this situation

Dont be confused: the Z0 in the expression of gamma_in is the load impedance, in the other case is the characteristic impedance

In your example, the Z0(61.24ohm) is the characteristic impedance of quarterwave transformer, so when you calculate the input reflection coefficient, you should take the Z0 as a 50ohm(source impedance).
As you say, the Zin=50 and the source impedance Z0 is 50ohm so the Gamma_in is zero.

When a source with source impedance Z1 feeds a load Z2, the gamma is
(Z1-Z2)/(Z1+Z2).

In your case, Z1=Zin=50 and Z2=50 (as the load ZL which is 75 is now reflected as 50 ohm at the source end). Therefore gamma is 0.