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I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovisky (BV) or antifield formalism:
$\rightarrow$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\rightarrow$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovisky (BV) or antifield formalism:
$\rightarrow$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\rightarrow$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.

Edit: The question: What is the BRST-anti-BRST formalism? and the answer there, are also related and may be of some further help.

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovinsky (BV) or antifield formalism:
$\rightarrow$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\rightarrow$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovisky (BV) or antifield formalism:
$\rightarrow$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\rightarrow$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\circ$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovinsky (BV) or antifield formalism:
$\circ$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\circ$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.

I do not think that it makes sense to say that the gauge-fixing is a special case of BRST quantization:

$\rightarrow$ The gauge-fixing procedure is actually a normalization technique and it is utilized in order to eliminate unphysical degrees of freedom (gauge directions) in the path integral quantization of the various gauge theories. This -rather naturally- destroys the gauge invariance. A standard method to restore the lost gauge invariance -and thus to recover the physics- is the Batalin-Vilkovinsky (BV) or antifield formalism:
$\rightarrow$ after introducing some extra fields in the theory (ghost fields and conjugate antifields), an extended action is constructed involving all these variables. This action is invariant under the BRST symmetry operator $s$ (usually called BRST-differential), which now replaces the original gauge symmetry. Its cohomology essentially contains the physics, in the following sense:
$\rightarrow$ The BRST-differential is nilpotent $s^2=0$ and thus its cohomological groups $H^n(s)$ can be constructed. $H^0(s)$ consists of the gauge invariant functions (that is: the observables).

You can find more details in these notes and the references therein.