Dejar is used to talk about a object "voy a dejar mi mochila en casa" "i'm going to leave my backpack at home". And quedar to talk about person " nos quedaremos en casa" "we wil stay at home". Also you can use quedar to talk about agreement " quede a cenar con Jessica" "i organized a dinner with jessica" and dejar to finish a acction " dejaré de comer" "i'm gonna stop to eat" , this way are informal expressions.

Dejar is to leave behind or to stop an action, quedar is usually just referring to staying.

Voy a dejar de hablar sobre esas cosas = I'm gonna stop talking about that stuff

Te voy a dejar un mensaje = I'm gonna leave you a message (voicemail)

Te vas a quedar en tu casa? = Are you gonna stay at home?

Me voy a quedar callado = I'm just gonna stay shut

Dejar and quedar are both used in multiple situations, so more context would be needed to explain the differences between both verbs in specific phrases.

Anyway, they're nothing alike. Dejar means "let", "allow", "leave", "abandon", "lend"... while quedar means "remain", "stay", "have an appointment"...