Set the gridlinestyle property.
set(gca,'gridlinestyle','-')
As Matt Tearle has pointed out, you cannot change the grid lines color independent from the axis color. If you really want that, you should use colored lines instead of grids. For example:
plot(1:10)
axis([1 10 1 10]);
xt = get(gca,'xtick');
xl = get(gca,'xlim');
line(repmat(xt(2:end-1),2,1),repmat(xl(:),1,length(xt)-2),'color','red')
yt = get(gca,'ytick');
yl = get(gca,'ylim');
line(repmat(yl(:),1,length(yt)-2),repmat(yt(2:end-1),2,1),'color','g')
