The First Sino-Japanese War (1 August 1894 – 17 April 1895) was fought between the Qing Dynasty of China and the Empire of Japan, primarily over influence of Korea. In that case, demands might be made, or both nations may agree to a 'white peace', which is a return to the status quo ante bellum (the prewar situation). Pressing for peace may sometimes, however, be started by the winning faction as a means to end the war for several reasons, such as if additional conflict would not be in the perceived best interest of the winning party. Therefore, in this usage, the verb 'to sue' means 'to ask' or 'to seek' rather than 'to take legal action against'. The nation holding the upper hand may find, in the losing party’s offer of making peace, an opportunity for relief from the necessity of having to continue to wage a costly war. Suing for peace is usually initiated by the losing party in an attempt to stave off an unconditional surrender.
'Suing for', in this older sense of the phrase, means 'pleading or petitioning for'.