No, and for a variety of reasons. The most significant reason is that the gravitational field of the Sun and the Milky Way are stronger than the local gravitational field of the universe. This means that the dynamics of spacetime in our solar system and Milky Way are dominated by the curvature produced by these local masses. As an example, if the cosmological expansion could be detected in our solar system, its 60 kilometers/sec/megaparsecs would translate into a space dilation rate of 6 centimeters per second per parsec, or for a scale inside our solar system, 0.0002 centimeters/sec per billion kilometers. In 100 years this stretching would amount to 6.2 kilometers at the solar system scale, and 186,000 kilometers at the interstellar scale. Neither of these are measurable, nor is there any physical reason from general relativity why they should even be present given the strength of the local sources of gravity which completely overpower the effect.