Try

f[x_]:=(a x^2+b x+c)/(d x^2+e x+m);
FindInstance[{D[f[x1],x1]==0,D[f[x2],x2]==0,
  0<a<=1,0<b<=1,0<c<=1,0<d<=1,0<e<=1,0<m<=1,
  x1!=x2,Element[x1|x2,Integers]},{a,b,c,d,e,m,x1,x2}]

which finds an instance which seems to satisfy your conditions.

Try

f[x_]:=(a x^2+b x+c)/(d x^2+e x+m);
FindInstance[{D[f[x1],x1]==0,D[f[x2],x2]==0,
  0<a<=1,0<b<=1,0<c<=1,0<d<=1,0<e<=1,0<m<=1,
  x1!=x2,Element[x1|x2,Integers]},{a,b,c,d,e,m,x1,x2}]

which finds an instance which seems to satisfy your conditions.

If you change all those <= to < then it finds a different solution.