You can achieve that using TreeView expanded prop.
The code below expands the TreeItem with id "1" on mount.

import React, {useEffect, useState} from 'react';
import TreeView from '@material-ui/lab/TreeView';
import ExpandMoreIcon from '@material-ui/icons/ExpandMore';
import ChevronRightIcon from '@material-ui/icons/ChevronRight';
import TreeItem from '@material-ui/lab/TreeItem';

export default function FileSystemNavigator() {
  const [expanded, setExpanded] = useState([]);

  useEffect(() => {
    setExpanded(["1"])
  },[])

  return (
    <TreeView
      defaultCollapseIcon={<ExpandMoreIcon />}
      defaultExpandIcon={<ChevronRightIcon />}
      expanded={expanded}
    >
      <TreeItem nodeId="1" label="Applications"    
>
        <TreeItem nodeId="2" label="Calendar" />
        <TreeItem nodeId="3" label="Chrome" />
        <TreeItem nodeId="4" label="Webstorm" />
      </TreeItem>
      <TreeItem nodeId="5" label="Documents"
>
        <TreeItem nodeId="6" label="Material-UI">
          <TreeItem nodeId="7" label="src">
            <TreeItem nodeId="8" label="index.js" />
            <TreeItem nodeId="9" label="tree-view.js" />
          </TreeItem>
        </TreeItem>
      </TreeItem>
    </TreeView>
  );
}

Code Sandbox

The TreeView is "uncontrolled" by default meaning that the component will handle the state for you, and you as a consumer get whatever default behavior MUI set. In order to achieve what you want you'll need to make the TreeView a "controlled" component. There is an example of using the controlled component here: https://mui.com/components/tree-view/#controlled-tree-view

The example on the MUI page is doing more than just handling the expanding/collapsing of nodes. In the simplest form you need to explicitly handle the "expanded" prop that gets passed into the TreeView yourself.

const Tree = () => {
  // nodes "1", "1.1", "2" will start expanded
  const [expanded,setExpanded] = useState(["1","1.2","2"]);

  const createHandler = (id) => () => {
    // if node was collapsed, add to expanded list
    if(!expanded.includes(id)) {
      setExpanded([...expanded,id]);
    } else {
      // remove clicked node, and children of clicked node from expanded list
      setExpanded(expanded.filter(i => i !== id && !i.startsWith(`${id}.`));
    }
  };

  return (
    <TreeView expanded={expanded}>
      <TreeItem nodeId="1" label="A" onClick={createHandler("1")}>
        <TreeItem nodeId="1.1" label="B" />
        <TreeItem nodeId="1.2" label="C" onClick={createHandler("1.2")}>
          <TreeItem nodeId="1.2.1" label="D" />
        </TreeItem>
      </TreeItem>
      <TreeItem nodeId="2" label="E" onClick={createHandler("2")}>
        <TreeItem nodeId="2.1" label="F" />
      </TreeItem>
    </TreeView>
  );
}

Note: there be some mistakes in the above example, I'm unable to run it at the moment to verify correctness.

Probably you didn't set the callback when you try to control it. (Same as TextField's value and onChange)
TreeView API document here
You can find there the callback onNodeToggle
Set it will fix this problem.

To accomplish this, you need to override the default behavior of the TreeView component. This isn't extremely difficult, but can be somewhat confusing because the documentation isn't great.

You will have to make the expanded property of the TreeView controlled. Basically, this means having a state control which nodes are expanded/collapsed.

There are multiple ways to interact with the node, there are onClick methods, such as the onNodeSelect, onNodeToggle, and etc., you will need a way of capturing when the user clicks-to-expand.

Once you have the above, you can just have a function modify the state like so:

import { TreeItem, TreeView } from "@mui/lab";

function TreeDiagram(): JSX.Element {
   const [expanded, setExpanded] = React.useState<string[]>([]);
   

   /**
    * Method to remove or include a node to the array of expanded node
    */
   const handleExpandedToggle = (nodeId: string): void => {
      setExpanded((prevExpandedState: string[]): string[] => 
         prevExpandedState.includes(nodeId)
            ? prevExpandedState.filter((node: string) => node !== nodeId)
            : [nodeId, ...prevExpandedState]);
   };

   /**
    * Method to handle the click event of the TreeItem's
    */
   const handleTreeItemClick = (event: React.SyntheticEvent, nodeId: string): void => {
      handleExpandedToggle(nodeId);
   };

   return (
      <TreeView
         expanded={expanded}
      >
         {/* All the tree items */}
         <TreeItem
            onNodeFocus={handleTreeItemClick}
            nodeId={"tree_item_1"}
            label={ /* Whatever you'd like here */ }
         ></TreeItem>
      </TreeView>
   );
};