It seems that it needs #include <cairo/cairo-xlib.h>.

libgitg/gitg-platform-support.c:76:7: error: implicit declaration of function 'cairo_xlib_surface_get_width' is invalid in C99
      [-Werror,-Wimplicit-function-declaration]
                w = cairo_xlib_surface_get_width (surface);
                    ^
libgitg/gitg-platform-support.c:76:7: note: did you mean 'cairo_image_surface_get_width'?
/usr/local/include/cairo/cairo.h:2537:1: note: 'cairo_image_surface_get_width' declared here
cairo_image_surface_get_width (cairo_surface_t *surface);
^
libgitg/gitg-platform-support.c:77:7: error: implicit declaration of function 'cairo_xlib_surface_get_height' is invalid in C99
      [-Werror,-Wimplicit-function-declaration]
                h = cairo_xlib_surface_get_height (surface);
                    ^
2 errors generated.