You need to step away from the word "becomes" and maybe rethink your domain language.
In the code a Call object can't "become" a Quote object. My guess would be you create a new Quote based on the data from the Call and persist both of them to a db with a relationship. Maybe a shared key.
You can UML that up no problem because they are all operations ...
The UML sequence diagram is well adapted to represent an interaction between several objects (or components or, why not,systems). It has semantic to make the difference between synchronous and asynchronous communication. You may even show delay in message passing (but don't abuse of it).
However for a more complex interactions between systems, you should ...
A line without an arrow (usually a dashed line) is a trust boundary and is usually drawn over and perpendicular to one or more arrowed lines. For example, it may indicate that entities on one side are on a web page and ones on the other are on the server. This is not a part of the traditional data flow diagrams but is used in threat modelling. See the STRIDE ...
Just looking at the diagram without looking at the code:
the text over edges outgoing from a decision node should always be [ between bracket ]
the decision making cascade with several [ none of the above ] and check for more cases seems seems weird. We should be able to make th choice not knowing the history of the flow:
use [ else ] guard instead of [...
The main point of a diagram is to convey information. Exactly how it does so is usually a secondary concern. UML and other common types of diagram give us a shared diagrammatic language, but there's no reason you can't invent a type of diagram, as long as it says what you mean.
I'm not aware of any standard diagram format that conveys what you want out-of-...
An entity within the system that transforms data. Includes, for example, accounting clerks (persons), departments (places), and computers (things)
Source: Gelinas, Ulric J., et al. Business Processes and Information Technology. Orange Grove Texts Plus, 2008.
What do you expect from the diagram ?
To be very provocative: What does your diagram tell you that you do not already know ? Isn't this diagram confusing since Urdu Boli the game and everybody will wonder about its relation with the Game System ?
Also note that you foresee data to be saved, but at no place do you intend to reload this data.
Just some additional remarks to what Christophe said:
The "Step n" prefix seems weird since half of the actions don't have that.
throws means an exception. You'd put that in an interruptible region and exit with an InterruptFlow (that flashy arrow). See bottom of https://www.uml-diagrams.org/activity-diagrams.html