User-friendly single-photon sources with high photon-extraction efficiency are crucial building blocks for photonic quantum applications. For many of these applications, such as long-distance quantum key distribution, the use of single-mode optical fibers is mandatory, which leads to stringent requirements regarding the device design and fabrication. We report on the on-chip integration of a quantum dot microlens with a 3D-printed micro-objective in combination with a single-mode on-chip fiber coupler. The practical quantum device is realized by deterministic fabrication of the QD-microlens via in-situ electron-beam lithography and 3D two-photon laser writing of the on-chip micro-objective and fiber-holder. The QD with microlens is an efficient single-photon source, whose emission is collimated by the on-chip micro-objective. A second polymer microlens is located at the end facet of the single-mode fiber and ensures that the collimated light is efficiently coupled into the fiber core. For this purpose, the fiber is placed in the on-chip fiber chuck, which is precisely aligned to the QD-microlens thanks to the sub-$mu$m processing accuracy of high-resolution two-photon direct laser writing. This way, we obtain a fully integrated high-quality quantum device with broadband photon extraction efficiency, a single-mode fiber-coupling efficiency of 26%, a single-photon flux of 1.5 MHz at single-mode fibre output and a multi-photon probability of 13 % under pulsed optical excitation. In addition, the stable design of the developed fiber-coupled quantum device makes it highly attractive for integration into user-friendly plug-and-play quantum applications.