I followed this tutorial so I have a network with two wireless interfaces, wlan0 working as a access point and wlan1 working as a client from another router. How can I bridge wlan1 to eth0 without installing any extra software if possible? I'm using raspbian buster.
You are asking how to bridge eth0 with the uplink connection wlan1. This is not an easy task because bridging a client connection can only be done with WDS mode and must be supported by the hardware. The built-in device of the Raspberry Pi does not support it. For further information about this issue you can look at Raspberry Pi WiFi to Ethernet Bridge for a server?. But you could use the USB/WiFi dongle if it supports WDS.
But according to the clarification in your comment this does not solve your problem. To solve it you have to bridge eth0 with the access point interface wlan0. This is not a problem and can easily be done by following setup Using an additional WiFi dongle to create WAP and switch.