It is appropriate for believers to want to serve God forever because He has so graciously gifted us with eternal salvation. Indeed, the Apostle John confirms that we will serve God and Jesus forever in the New Jerusalem:
The throne of God and of the Lamb will be in the city, and his servants will serve him. 4 They will see his face, and his name will be on their foreheads. 5 There will be no more night. They will not need the light of a lamp or the light of the sun, for the Lord God will give them light. And they will reign for ever and ever. (Revelation 22:3b-5).
Yes, we will serve Him for ever and He will put “His name” on our “foreheads” to show that we are His intimate property. God will be our light and our lamp, and somehow, we will “reign” forever, even though we are servants. I don’t understand it all, but I look forward to it, because I will spend eternity with the One who loves me most of all.