QuerySelector of LWC Tree Grid Row

You can't peer past the Shadow DOM, so you're only allowed to style the "outside" of the component. You won't be able to affect elements inside the tree either by CSS or by JS, except as outlined in Styling Hooks. You'd need to make your own component, instead.

Ultimately, I decided to highlight the target row by simply selecting it. The implementation of this component already hides checkboxes from the user, so this was a very simple way to identify the row. Unfortunately, there are no styling hooks for changing the selection color, so I also added an icon.

// component.html
<template>
  <div class="slds-p-around_medium">
    <template if:true={gridData}>
      <lightning-tree-grid
        columns={gridColumns}
        data={gridData}
        key-field="Id"
        is-loading={isLoading}
        hide-checkbox-column=true
        ontoggle={handleRowToggle}
        expanded-rows={currentExpanded}
        show-row-number-column=true
      >
      </lightning-tree-grid>
    </template>
  </div>
</template>

// component.js (edited for brevity)
import { LightningElement, api, track } from 'lwc';
import getInitialData from '@salesforce/apex/ComponentController.getInitialData';
import getChildRecords from '@salesforce/apex/ComponentController.getChildRecords';

export default class Component extends LightningElement {
  @api recordId;
  isLoading = false;
  selectedRow = [];
  @track gridData;
  @track gridColumns = [
    {
      type: 'url',
      fieldName: 'url',
      label: 'Name',
      typeAttributes: {
        label: { fieldName: 'Name' },
        tooltip: { fieldName: 'Name' },
      },
      cellAttributes: {
        iconName: { fieldName: 'isCurrentAccount' },
        iconPosition: 'right',
        iconAlternativeText: 'Current Account',
      },
    },
    // other columns
];

  connectedCallback() {
    this.fetchRecords(this.recordId, true);
  }

  fetchRecords(parentRecordId, onLoad = false) {
    this.isLoading = true;
    this.getData(onLoad)({
      recordId: parentRecordId,
    })
    .then((serverData) => {
      if (serverData) {
        const newGridData = serverData.map(record => this.buildTreeNode(record));

        if (onLoad) {
          this.gridData = newGridData;
          this.selectedRow = [this.recordId]; // SELECT THE TARGET RECORD HERE
        } else {
          // async loading of nested records
        }

        this.isLoading = false;
      }
    })
    .catch((error) => {
      // handle errors
    });
  }

  buildTreeNode(element) {
    let record = element.record;

    // SPECIFY THE ICON VALUE HERE
    if (record.Id === this.recordId) {
      record.isCurrentAccount = 'utility:success';
    }

    // other stuff, mostly related to the nested _children

    return record;
  }

  // the rest, unrelated to this question
}

You can use a utility class to change the background but options are limited list of utility classes. You can bind the class to a grid node property and only define that property on the row you want to highlight (see above code of OP for context):

if (record.Id === this.recordId){
    
   node.currentAccountClass = 'slds-theme slds-theme_success';

}

and in your column definition you can use the following:

cellAttributes: {
                class: { fieldName: 'currentAccountClass' } 
            }