AngularDegrees^2 and Steradians are incompatible units

It seems you have misspelled "Degrees". When I correct your misspelling, I get

UnitConvert[Quantity[360, "Degrees^2"], "Steradians"]

Quantity[Pi^2/90, "Steradians"]

It quite a bit less than $4 \pi$. To find the angular measure of a complete sphere in angular degrees we can do the following:

UnitConvert[Quantity[4 π, "Steradians"], "Degrees^2"]

Quantity[129600/π, ("AngularDegrees")^2]

Also, note that a steradian is the same as a square radian.

UnitConvert[Quantity[1, "Steradians"], "Radians^2"]

Quantity[1, "Radians"^2]

We can't seem to reproduce your problem.

Here are the results of your simpler question from comments, first on my laptop:

$Version
UnitConvert[Quantity[1, "Steradians"], "Radians^2"]

(* Out: 
"12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)"
Quantity[1, "Radians"^2] 
*)

... and on the cloud version:

$Version
UnitConvert[Quantity[1, "Steradians"], "Radians^2"]

(* Out: 
"12.2.0 for Linux x86 (64-bit) (November 16, 2020)"
Quantity[1, "Radians"^2]
*)