Is it possible to use material-ui button navigation with react-router?

Yes, it's possible. You need to use the component prop:

import { Link } from 'react-router-dom';

import BottomNavigation from '@material-ui/core/BottomNavigation';
import BottomNavigationAction from '@material-ui/core/BottomNavigationAction';

// ....

<BottomNavigation value={value} onChange={this.handleChange}>
    <BottomNavigationAction
        component={Link}
        to="/signal"
        label="signal"
        value="signal"
        icon={<ShowChart />}
        className={classes.content}
    />
</BottomNavigation>

(the to prop is for React Router's Link component)

This works with any Material-UI component that inherits from ButtonBase.

https://material-ui.com/api/bottom-navigation-action/

Inheritance

The properties of the ButtonBase component are also available. You can take advantage of this behavior to target nested components.

https://material-ui.com/api/button-base/

Props

component - The component used for the root node. Either a string to use a DOM element or a component.

Just to add on to the great answer by @thirtydot , in case the user types into the search and visits a particular webpage directly (other than default) e.g. "www.yoursite.com/tab2", instead of clicking the 2nd button, this may cause a mismatch between the site that is showing and the BottomNav Button that is focused (usually the 1st button).

Here is what I did:
I used window.location.pathname to get the current path which is '/tab2' directly.
Here is my code for my particular use case....

function BottomNavBar(){
    const pathname = window.location.pathname; // in case user visits the path directly. The BottomNavBar is able to follow suit.
    const [value, setValue] = React.useState(pathname);
    const handleChange = (event, newValue) => {
      setValue(newValue);
    };

    return (
        <BottomNavigation value={value} onChange={handleChange} showLabels={true} >
          <BottomNavigationAction label="home" value="/" icon={<HomeIcon />} component={Link} to='/'/>
          <BottomNavigationAction label="resources" value="/resources" icon={<ResourcesIcon /> } component={Link} to='/resources'/>                
          <BottomNavigationAction label="Q&A" value="/qna" icon={<QnAIcon />}  component={Link} to='/qna'/>
          <BottomNavigationAction label="profile" value="/profile" icon={<ProfileIcon />} component={Link} to='/profile'/>
        </BottomNavigation>
      );
}