Berakhot 7a discusses the topic of God’s prayer:
R. Johanan says in the name of R. Jose: How do we know that the Holy One, blessed be He, says prayers? Because it says: Even them will I bring to My holy mountain and make them joyful in My house of prayer (Isaiah 56:7; more below). It is not said, ‘their prayer’, but ‘My prayer’ [תְּפִלָּתִי]; hence [you learn] that the Holy One, blessed be He, says prayers.
What does He pray? — R. Zutra b. Tobi said in the name of Rab: ‘May it be My will that My mercy may suppress My anger, and that My mercy may prevail over My [other] attributes, so that I may deal with My children in the attribute of mercy and, on their behalf, stop short of the limit of strict justice.’…
Isaiah 56:7
וַהֲבִיאוֹתִים אֶל-הַר קָדְשִׁי, וְשִׂמַּחְתִּים בְּבֵית תְּפִלָּתִי–עוֹלֹתֵיהֶם וְזִבְחֵיהֶם לְרָצוֹן, עַל-מִזְבְּחִי: כִּי בֵיתִי, בֵּית-תְּפִלָּה יִקָּרֵא לְכָל-הָעַמִּים
Even them will I bring to My holy mountain, and make them joyful in My house of prayer; their burnt-offerings and their sacrifices shall be acceptable upon Mine altar; for My house shall be called a house of prayer for all peoples.
–“Old” JPS translation (borrowed from Mechon-Mamre)
"A Song Every Day" 