Update google search url for QSE

[deeppandya]

In this issue, we need to update google search url for quick search engine.
Below is the url we get from template URL service for google service,
{google:baseURL}search?q={searchTerms}&{google:RLZ}{google:originalQueryForSuggestion}{google:assistedQueryStats}{google:searchFieldtrialParameter}{google:language}{google:prefetchSource}{google:searchClient}{google:sourceId}{google:contextualSearchVersion}ie={inputEncoding}
we need to update it with : https://www.google.com/search?q={searchTerms} for google serach

[kjozwiak]

The above requires 1.74.13 or higher for 1.74.x verification 👍 Labelled as QA/Yes as the above should be checked. @Uni-verse went through the above on Nightly and added some STR/Cases that can be used for 1.74.x verification via brave/brave-core#26550 (comment).

[hffvld]

Verified on Pixel 7 using version(s):

Device/OS: Pixel 7 / panther_beta-user 15 AP41.240925.009 release-keys
Brave build: 1.74.13 
Chromium: 131.0.6778.85 (Official Build) beta (64-bit) 

Filed follow-up issue #42565

Fresh profile

STEPS:

1. Fresh install Brave Beta 1.74.13 or higher
2. Go through onboarding
3. Open NTP
4. Tap in the URL search bar
5. Type a query > Verify that QSE toolbar is shown
6. Tap Google search thumbnail > Verify that the search results are shown in Google https://www.google.com/search?q={searchTerms}

ACTUAL RESULTS:

• Verified that QSE is not shown until the user starts typing a query in the URL search bar.
• Verified that the Google search works when tapping the thumbnail.
• Verified that the results are shown with https://www.google.com/search?q={searchTerms}.

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2024-11-27_11-32-22.mp4

Upgraded profile

STEPS:

1. Install Brave Beta 1.74.10 (it was previously available in the Play Store)
2. Go through onboarding
3. Open NTP
4. Tap in the URL search bar > Type a query
5. Tap Google search thumbnail > Observe that Google search doesn't work
6. Upgrade Brave to Beta 1.74.13 or higher
7. Type a query again
8. Tap Google search thumbnail > Observe that Google search is still not working

ACTUAL RESULTS:

• Verified that QSE is not shown until the user starts typing a query in the URL search bar after Brave upgraded to a new build.
• Observed that the Google search didn't work even after Brave upgraded to the new build.

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2024-11-27_11-21-12.mp4