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.