It depends a bit how you are going to need that state. If the object is an output from one activity function and passed into another then you can use the function chaining pattern such as Sajeetharan described because the inputs for the activity function are persisted by default.
But if you want to capture the state of an object over a longer time (i.e. the last time your timer function ran) you can look into Durable Entities (still in preview at the time of writing):
Entity functions define operations for reading and updating small
pieces of state, known as durable entities. Like orchestrator
functions, entity functions are functions with a special trigger type,
entity trigger. Unlike orchestrator functions, entity functions do not
have any specific code constraints. Entity functions also manage state
explicitly rather than implicitly representing state via control flow.
Chris Gillum wrote an article about it: https://medium.com/@cgillum/azure-functions-durable-entities-67db648d2f74
If you can't use this preview version or don't like concept you can still write activity functions that have input & output bindings to table- or blob storage or CosmosDB and manage the state yourself.