Using Bioalcidae and the following script?

var counts={};
while(iter.hasNext()) {
var ctx  = iter.next();
for(var i=0;i< ctx.getNSamples();++i)
    {
    var gi = ctx.getGenotype(i);
    for(var j=i+1;j< ctx.getNSamples();++j)
        {
        var gj = ctx.getGenotype(j);
        if(  gi.sameGenotype(gj) ) {
            var key = gi.getSampleName()+"-"+gj.getSampleName();
            var c= counts[key];
            if(c==null) c=0;
            ++c;
            counts[key]=c;
            }
        }
    }
}

for(var i in counts) {
    out.println("["+i+"]\t"+counts[i]);
    }

run:

gunzip -c input.vcf.gz| java -jar dist-2.0.1/bioalcidae.jar -F vcf -f script.js | column -t

[S1-S10]  35
[S1-S2]   16
[S1-S3]   20
[S10-S2]  16
[S10-S3]  20
[S2-S3]   19
[S1-S4]   16
[S10-S4]  16
[S3-S4]   19
[S2-S4]   15