Results of the first principles study on the electronic structure and magnetism of the superconducting weak ferromagnet Y$_4$Co$_3$, are presented. Using the full potential Korringa-Kohn-Rostoker (FP-KKR) method, densities of states, dispersion curves and magnetic moments were calculated for quasi-ordered structural model of the compound in the framework of the local density approximation. Spin-polarized KKR calculations confirm that weak ferromagnetic properties of Y$_4$Co$_3$ can be attributed to only one cobalt atom located on (2b) site in the unit cell, while other twenty Co and Y atoms acts as a diamagnetic environment. Moreover, the magnetic Co atoms form a quasi-one-dimensional chains along $z$ direction. The magnitude of Co(2b) magnetic moment ($0.55~\mu_B$) markedly overestimates the experimental value (0.23~$\mu_B$), which suggests the importance of spin fluctuations in this system. Calculated distribution of spin magnetization in the unit cell provides a background for discussion of the coexistence of ferromagnetism and superconductivity in Y$_4$Co$_3$. Finally, the effect of pressure on magnetism is discussed and compared with experimental data, also supporting weak ferromagnetic behaviors in the system.