Seems with Windows Phone 8.1 GDR1, the “Google” search option will disappear from the browsers. Till now, one can go to advanced settings of Internet explorer and set “Google” or “Bing” as search providers.
But it seems, If the device is running Windows Phone 8.1 GDR1, then it has to use “Bing” as the only search provider. Though, there are few markets including Russia, where “Yandex” will be used in place of Bing. Also, for markets in which Bing is not supported and a specific search provider is not required, partners can add a prioritized list of up to two additional search providers.
From the documentation,
By default, the only search provider included on the phone is Bing. The search provider used in the browser is always the same as the one launched by the hardware search button. However, in certain markets OEMs must specify a different search provider as the default search provider for Windows Phone. For more information about the default search provider that OEMs must set for these markets, see the section Default search providers for specific markets.
The phone automatically determines whether Bing is supported based on the browser and search language. This language is derived from the default region format, and can be modified by the user. If the phone’s browser and search language are set to a language that Bing supports, any search providers specified by the OEM will be hidden from the Internet Explorer screen in Settings, and both the hardware search button and search in the browser will always use Bing. Partners cannot control or change this determination. Windows Phone reserves the right to update the list of supported languages, and therefore override or hide custom OEM search providers, at any time.
About Yandex and unsupported markets,
For phones sold in Armenia, Azerbaijan, Belarus, Georgia, Kazakhstan, Kyrgyzstan, Moldova, Russia, Tajikistan, Turkey, Turkmenistan, Ukraine, and Uzbekistan, the following limitations and restrictions apply.
Yandex will provide an optional search application which OEMs can preload and configure on phones sold in these markets.
For markets in which Bing is not supported and a specific search provider is not required, partners can add a prioritized list of up to two additional search providers—these are listed on the Internet Explorer screen in Settings so that users can select a new search provider instead of Bing